43. use the diagram to the right to answer the questions below.
a. if ( mangle1 = 84^{circ} ), what must be ( mangle5 ) in order for ( aparallel b )?
b. if ( mangle2 = 109^{circ} ), what must be ( mangle9 ) in order for ( aparallel c )?
c. if ( mangle5 = 68^{circ} ), what must be ( mangle11 ) in order for ( bparallel c )?
d. if ( mangle3 = mangle9 ), what converse proves ( aparallel c )?
e. if ( mangle8 = mangle12 ), what converse proves ( bparallel c )?
f. if ( mangle2 + mangle5 = 180^{circ} ), what converse proves ( aparallel b )?

Question

Accepted Answer

### Part a
# Explanation:
## Step1: Identify Angle Relationship
For $a \parallel b$, $\angle1$ and $\angle5$ are corresponding angles. Corresponding angles are equal when lines are parallel.
## Step2: Determine $m\angle5$
Given $m\angle1 = 84^\circ$, so $m\angle5 = m\angle1 = 84^\circ$.
# Answer: $84^\circ$

### Part b
# Explanation:
## Step1: Identify Angle Relationship
For $a \parallel c$, $\angle2$ and $\angle9$ are corresponding angles. Corresponding angles are equal when lines are parallel.
## Step2: Determine $m\angle9$
Given $m\angle2 = 109^\circ$, so $m\angle9 = m\angle2 = 109^\circ$.
# Answer: $109^\circ$

### Part c
# Explanation:
## Step1: Identify Angle Relationship
For $b \parallel c$, $\angle5$ and $\angle11$ are corresponding angles. Corresponding angles are equal when lines are parallel.
## Step2: Determine $m\angle11$
Given $m\angle5 = 68^\circ$, so $m\angle11 = m\angle5 = 68^\circ$.
# Answer: $68^\circ$

### Part d
# Explanation:
## Step1: Identify Angle Relationship
$\angle3$ and $\angle9$ are corresponding angles. The Converse of Corresponding Angles Theorem states that if two lines are cut by a transversal and the corresponding angles are equal, then the lines are parallel.
## Step2: Apply the Converse
Since $m\angle3 = m\angle9$ (corresponding angles equal), the Converse of Corresponding Angles Theorem proves $a \parallel c$.
# Answer: Converse of Corresponding Angles Theorem

### Part e
# Explanation:
## Step1: Identify Angle Relationship
$\angle8$ and $\angle12$ are alternate interior angles. The Converse of Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior angles are equal, then the lines are parallel.
## Step2: Apply the Converse
Since $m\angle8 = m\angle12$ (alternate interior angles equal), the Converse of Alternate Interior Angles Theorem proves $b \parallel c$.
# Answer: Converse of Alternate Interior Angles Theorem

### Part f
# Explanation:
## Step1: Identify Angle Relationship
$\angle2$ and $\angle5$ are same - side interior angles. The Converse of Same - Side Interior Angles Theorem states that if two lines are cut by a transversal and the same - side interior angles are supplementary (sum to $180^\circ$), then the lines are parallel.
## Step2: Apply the Converse
Since $m\angle2 + m\angle5 = 180^\circ$ (same - side interior angles supplementary), the Converse of Same - Side Interior Angles Theorem proves $a \parallel b$.
# Answer: Converse of Same - Side Interior Angles Theorem

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part a

Step1: Identify Angle Relationship

Step2: Determine \(m\angle5\)

Step1: Identify Angle Relationship

Step2: Determine \(m\angle9\)

Step1: Identify Angle Relationship

Step2: Determine \(m\angle11\)

Answer:

Part b